PairConnect: Power of Pairwise Interactions

• PairConnect is a suite of models and methods that rigorously captures pairwise interactions, serving as a foundational unit of structure in diverse scientific domains.
• It leverages targeted pairwise methodologies to improve computational accuracy and efficiency, as seen in quantum many-body theory, machine learning, and network science.
• The framework enables practical innovations such as enhanced Monte Carlo sampling, robust connectomics classification, and advanced cryptographic pair mappings.

PairConnect refers to a suite of models, methods, and analytical frameworks unified by their focus on pairwise interactions and pairwise structure—spanning domains from quantum many-body theory and condensed matter physics to computational chemistry, neuroscience, number theory, machine learning, statistical mechanics, cosmology, and cryptography. Across these diverse fields, the "connectivity" of pairs—fermion pairs, particle pairs, graph edge pairs, sequence interval pairs, connectome pairs—is both a fundamental unit of structure and a computational strategy for improved accuracy, efficiency, or expressivity.

1. Principles of Pairwise Interaction and Connectivity

Pairwise modeling is foundational in physical and mathematical sciences, where the basic unit of correlation, connection, or interaction is rigorously captured at the two-object or two-site level.

• In quantum many-body systems, pair-connectedness often embodies the dominant contribution to correlation energy (e.g., electron pairs in the BCS wave function or seniority-zero sector in coupled cluster theory) (Henderson et al., 2014, Henderson et al., 2015).
• In statistical mechanics, pair contact processes and diffusively coupled particle pairs drive phase transitions and critical behavior, with explicit two-species models used to disentangle solitary from aggregated dynamics (Deng et al., 2020).
• In combinatorics, discrepancy theory, and analytic number theory, pairwise spacings, pair correlations (e.g., Poissonian pair correlations, Montgomery's pair correlation conjecture), and low-discrepancy construction are central to understanding uniformity, randomness, and spectral statistics (Weiß, 2021, Goldston et al., 19 Mar 2025).
• In machine learning, direct pairwise embedding mechanisms, such as PairConnect MLP, replace computationally expensive all-vs-all dot products (as in Transformer attention) with an explicit representation of every pairwise relationship between tokens (Xu et al., 2021).
• In network science and connectomics, classification frameworks leveraging pairwise differences in structural connectomes achieve near-perfect discrimination between individuals (Petrov et al., 2017).

Pairwise methodologies are therefore central both in capturing the dominant physics/statistics, and, in many cases, in providing powerful algorithmic optimizations or new representational paradigms.

2. Pairwise Models in Quantum Many-Body Theory and Chemistry

A major research thrust has focused on developing computationally tractable models that rigorously or approximately capture ground and excited state pair correlations.

Coupled Cluster Doubles with Pairing

• The pair coupled cluster doubles (p-CCD) method restricts coupled cluster theory to double excitations that preserve fermion pairs. In the standard Hartree–Fock basis, the method is accurate up to the point of spontaneous symmetry breaking, but fails for strongly correlated or symmetry-broken states. Extension to a quasiparticle (BCS) basis ("PairConnect" in this domain) enables accurate and low-cost description of strongly correlated pairing regimes and smooth navigation across the symmetry-breaking phase transition at critical coupling, with computational complexity $\mathcal{O}(N^3)$ as a function of system size (Henderson et al., 2014).

Pair Extended Coupled Cluster Doubles (pECCD)

• The pECCD method extends pCCD to achieve doubly occupied configuration interaction (DOCI)–quality results for both energies and wave functions—even for strongly attractive and symmetry-broken regimes—at mean-field computational cost (Henderson et al., 2015). In pECCD, a central role is played by pair excitation and de-excitation operators acting within seniority-zero space. The approach accurately recovers both energy and density matrix properties by leveraging a double similarity transformation, and is particularly effective for static correlation in bond breaking or systems dominated by pairwise electron correlation.

Exactly Solvable Pairing Models

• Many-body Hamiltonians with explicit pairwise terms (e.g., reduced BCS pairing, extended multi-pair interactions) admit analytic solutions via Bethe ansatz or group-theoretical methods (e.g., $sp(4)$ algebraic shell model), producing highly accurate predictions for nuclear binding energies and fine structure in isotopic and isobaric analog states. The algebraic formalism also allows quantification of competition between like-particle and proton-neutron pairing, providing deep insight into the interplay of pair-connected subspaces in finite systems (Draayer et al., 2018).

3. Pairwise Interactions in Statistical Mechanics and Critical Phenomena

The explicit modeling of pairwise degrees of freedom provides clarity in universality class identification, critical exponent measurement, and simulation methodology.

• In pair contact process with diffusion (PCPD), the introduction of a two-species model (particles $B$ and pairs $A$) resolves longstanding ambiguities concerning which microscopic configurations drive the macroscopic phase transition. The explicit coupling (e.g., $B+B\to A$; $A\to A+B$; $A\to \varnothing$) allows direct correspondence with single- and pair-connected order parameters, and seed simulations of the $A$ species yield universal scaling exponents matching those of classic PCPD, establishing full dynamical equivalence (Deng et al., 2020).

4. Pairwise Correlation and Discrepancy in Sequence Analysis and Number Theory

Pairwise statistical analysis is central in the paper of uniform distribution, randomness, and spectral properties.

• For sequences $(x_n)$ in $[0,1)$, Poissonian pair correlations (PPC) are a measure of local randomness, quantified via

$$F_N(s) = \frac{1}{N} \#\{1 \leq l \neq m \leq N : \{x_l - x_m\} \leq s/N \} \rightarrow 2s$$

as $N \to \infty$. It is proven that PPC implies uniform distribution of $(x_n)$, and more generally, that low-discrepancy sequences (optimal in star-discrepancy rate) always have so-called $\alpha$-pair correlations for any $0<\alpha<1$, even if they do not reach the full PPC (the $\alpha=1$ case), typically due to a finite gap structure (Weiß, 2021). This links global uniformity to pairwise local statistics.

• In analytic number theory, Montgomery's Pair Correlation Conjecture describes the asymptotic statistics of spacings between the nontrivial zeros of the Riemann zeta function. The conjecture's validity not only predicts the same statistical law as a random matrix spectrum (sine kernel statistics) but, through delicate multiplicity and location arguments, implies that asymptotically 100% of zeros are simple and on the critical line—establishing a deep connection between pair statistics and the finer arithmetic structure of zeta zeros (Goldston et al., 19 Mar 2025).

5. Pairwise Frameworks in Machine Learning and Network Science

Recent advances have framed pairwise connectivity as either an explicit modeling device or as an approach to efficiently capture interaction structure.

Memory-Compute Tradeoff in Attention-Like Architectures

• PairConnect (in machine learning) is a compute-efficient alternative to Transformer self-attention. It replaces quadratic all-pairs dot products with explicit, learnable pairwise embeddings stored in a hash table, aggregated via a multilayer perceptron. This allows the model to learn arbitrary binary functions $F_{\text{memMap}}(x_i, x_j)$ representing interaction between tokens $x_i$ and $x_j$. The approach achieves comparable masked LLMing accuracy to Transformers (e.g., on PTB and WikiText-2 corpora), with up to 22% higher CPU inference throughput, and is proven strictly more expressive than dot product attention due to the lack of a structural factorization constraint (Xu et al., 2021).

Pairwise Classification of Structural Connectomes

• In connectomics, pairwise feature mappings and difference norm computations form the basis of robust classification algorithms distinguishing same-subject and different-subject connectomes with ROC AUC as high as 0.99. The method leverages both direct bag-of-edges features and rich node-level network metrics (e.g., PageRank, local efficiency) and functions as both a subject identifier and a filtering stage to remove non-individual-specific features in longitudinal brain studies (Petrov et al., 2017).

6. Improved Statistical Estimation and Monte Carlo via Pairwise Structure

Pairwise (quasi) Monte Carlo sampling and pair correlation can dramatically improve estimation accuracy for spatial statistics and correlation functions.

• In cosmological two-point correlation function estimation, replacing random point sets with low-discrepancy sequences (e.g., randomized Halton sequences) in pair-count-based estimators produces $1/N$ error scaling, compared to $1/\sqrt{N}$ for standard Monte Carlo, resulting in up to an order-of-magnitude more accurate results at the same computational cost using Corrfunc's optimized routines (Kerscher, 11 Jan 2024).
• This methodology is also supported by analytic results connecting low-discrepancy structure, pair correlation statistics, and error bounds in various probabilistic and combinatorial settings (Weiß, 2021).

7. Pairwise Maps in Cryptography and Multilinear Generalizations

Pairwise mapping is a central algebraic operation in cryptography.

• Bilinear pairings $e : G_1 \times G_2 \to G_T$ on elliptic curves, efficiently computable via Miller’s algorithm, are foundational in constructing identity-based encryption, short and aggregate signatures, and non-interactive key exchange protocols. The core bilinearity property $e(aP,bQ) = e(P,Q)^{ab}$ enables advanced protocol designs, though at significant computational cost compared to standard EC operations.
• Research is ongoing into cryptographically robust multilinear (i.e., tri- and higher-order) extensions. While they promise even more flexible protocols (such as broadcast encryption or multi-party Diffie-HeLLMan), all known constructions make trade-offs between security assumptions, computational efficiency, and practicality (Kumar et al., 2021).

Concluding Perspective

PairConnect, across its instantiations, represents the methodological and conceptual crystallization of pairwise structure as a fundamental organizing principle for improved modeling, analysis, and computation. Rigorous advances in this area have led to significant methodological gains in fields as diverse as quantum chemistry, number theory, cosmology, machine learning, network science, statistical mechanics, and cryptography. The continuous evolution of pairwise frameworks—including generalizations to higher-order connections, optimized computational realizations, and exploration of universality—remains central to contemporary theoretical and applied research at the intersection of mathematics, physics, data science, and engineering.